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# [Smart Math] Ratio Proportion Problem 16

Here’s and example of a SMART MATH problem for RATIO PROPORTION.

### Problem

After A has finished 1/2 of a piece of work in 10 days; B joins him after 5 days. Later A completes the work in 4 more days. In how many days would B alone complete the work?

1. $7\frac{1}{2}$ days
2. $16\frac{2}{3}$ days
3. $33\frac{1}{3}$ days
4. 100 days
5. 120 days

### The Usual Method

If A finishes 1/2 of the work in 10 days, it would take 20 days for A to finish the work alone. Assume that B takes ‘2’ days to complete the work alone. Thus in 1 day, B will do $\frac{1}{b}$ of the work. So, in 5 days, B will finish $\frac{5}{b}$ of the work.

Work done by A in 5 days = $\frac{5}{20}=\frac{1}{4}$

Work done by A in 4 days = $\frac{4}{20}=\frac{1}{5}$

The work done by A & B in sequence as mentioned in the question is:

$\frac{1}{2}$ (by A) + $\frac{1}{4}$ (by A) + $\frac{5}{b}$ (by B) + $\frac{1}{5}$ (by A) = 1

Hence, $\frac{1}{2}$ + $\frac{1}{4}$ + $\frac{5}{b}$ + $\frac{1}{5}$ = 1

$\therefore \frac{10+5+4}{20}+\frac{5}{b}=1$

$\therefore \frac{19}{20}+\frac{5}{b}=\frac{20}{20}$

$\therefore \frac{5}{b}=\frac{1}{20}$

$\therefore b=100$days

(Ans: 4)

Estimated Time to arrive at the answer = 75 seconds.

### Using Technique

Presently A works for a total of 10 + 5 + 4 days, i.e 19 days. A alone can finish the work in 20 days. This means that contribution of B is equivalent to saving 1 days work of A or $\frac{1}{20}$ of the total work. B contributes this $\frac{1}{20}$ of the work by working for 5 days, so in 1 day, B will do $\frac{1}{100}$ of the work. Thus, B will take 100 days to finish the work alone.